A significant challenge in physics-informed ML architectures is to develop implementations that
incorporate known physics but also scale well to higher dimensions [1]}.
A major contribution of this work is several exact solutions, in terms of standard linear algebra operations, of
Procrustes problems of physical relevance.
Where possible, we have made use of matrix factorizations and low-rank representations to alleviate the computational burden of implementation;
it is rarely necessary to form \(A\)  explicitly when computing its spectral properties or other diagnostics.
For complicated matrix manifolds, an exact solution of the Procrustes
problem (REF ) may be intractable and we must resort to algorithmic approaches.
Fortunately, optimization on matrix manifolds is a mature field [2]},
and many algorithms are implemented in the open source
software package `Manopt' [3]}.
